Probability Theory Random Variables and Distributions

Rob Nicholls

MRC LMB Statistics Course 2014

Introduction

Introduction

Significance magazine (December 2013) Royal Statistical Society

Introduction

Significance magazine (December 2013) Royal Statistical Society

Introduction

Significance magazine (December 2013) Royal Statistical Society

Contents

•  Set Notation •  Intro to Probability Theory •  Random Variables •  Probability Mass Functions •  Common Discrete Distributions

Set Notation

A set is a collection of objects, written using curly brackets {}

If A is the set of all outcomes, then:

A set does not have to comprise the full number of outcomes E.g. if A is the set of dice outcomes no higher than three, then:

A = {heads, tails}

A = {one, two, three, four, five, six}

A = {one, two, three}

Set Notation

If A and B are sets, then:

!A

A∪B

A∩B

A \ B

∅

Complement – everything but A

Union (or)

Intersection (and)

Not

Empty Set

Set Notation

Venn Diagram:

A B

one

two, three six four five

Set Notation

Venn Diagram:

A B

A = {two, three, four, five}

one

two, three six four five

Set Notation

Venn Diagram:

A B

B = { four, five, six}

one

two, three six four five

Set Notation

Venn Diagram:

A B

A∩B = { four, five}

one

two, three six four five

Set Notation

Venn Diagram:

A B

A∪B = {two, three, four, five, six}

one

two, three six four five

Set Notation

Venn Diagram:

A B

(A∪B ") = {one}

one

two, three six four five

Set Notation

Venn Diagram:

A B

A \ B = {two, three}

one

two, three six four five

Set Notation

Venn Diagram:

A B

(A \ B !) = {one, four, five, six}

one

two, three six four five

Probability Theory

To consider Probabilities, we need:

1. Sample space: Ω

2. Event space:

3. Probability measure: P

Probability Theory

To consider Probabilities, we need:

1. Sample space: Ω - the set of all possible outcomes

Ω = {heads, tails}

Ω = {one, two, three, four, five, six}

Probability Theory

To consider Probabilities, we need:

2. Event space: - the set of all possible events

Ω = {heads, tails}Ω = {{heads, tails},{heads},{tails},∅}

Probability Theory

To consider Probabilities, we need:

3. Probability measure: P P must satisfy two axioms:

P :F→ [0,1]

P(Ω) =1

P(iAi ) = P(Ai )

i∑

Probability of any outcome is 1 (100% chance) If and only if are disjoint A1,A2,…

Probability Theory

To consider Probabilities, we need:

3. Probability measure: P P must satisfy two axioms:

P :F→ [0,1]

P(Ω) =1

P(iAi ) = P(Ai )

i∑

Probability of any outcome is 1 (100% chance) If and only if are disjoint A1,A2,…

P({one, two}) = P({one})+P({two})13=16+16

Probability Theory

To consider Probabilities, we need:

1. Sample space: Ω 2. Event space: 3. Probability measure: P

As such, a Probability Space is the triple: (Ω, ,P )

Probability Theory

To consider Probabilities, we need:

The triple: (Ω, ,P ) i.e. we need to know: 1.  The set of potential outcomes; 2.  The set of potential events that may occur; and 3.  The probabilities associated with occurrence of those events.

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

A = {one, two}!A = {three, four, five, six}

P(A) =1/ 3P( !A ) = 2 / 3

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A B

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A B

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A B

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A = {one, two}B = {two, three}A∪B = {one, two, three}A∩B = {two}

P(A) =1/ 3P(B) =1/ 3P(A∪B) =1/ 2P(A∩B) =1/ 6

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A ⊆ B P(A) ≤ P(B) P(B \ A) = P(B)−P(A)If then and

A B

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A ⊆ B P(A) ≤ P(B) P(B \ A) = P(B)−P(A)If then and

A B

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A ⊆ B P(A) ≤ P(B) P(B \ A) = P(B)−P(A)If then and

A B

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A ⊆ B P(A) ≤ P(B) P(B \ A) = P(B)−P(A)If then and

A B

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(A∪B) = P(A)+P(B)−P(A∩B)

A ⊆ B P(A) ≤ P(B) P(B \ A) = P(B)−P(A)If then and

A = {one, two}B = {one, two, three}B \ A = {three}

P(A) =1/ 3P(B) =1/ 2P(B \ A) =1/ 6

Probability Theory

Notable properties of a Probability Space (Ω, ,P ):

P( !A ) =1−P(A)

P(∅) = 0

P(A∪B) = P(A)+P(B)−P(A∩B)

A ⊆ B P(A) ≤ P(B) P(B \ A) = P(B)−P(A)If then and

Probability Theory So where’s this all going? These examples are trivial!

Probability Theory So where’s this all going? These examples are trivial! Suppose there are three bags, B1, B2 and B3, each of which contain a number of coloured balls: •  B1 – 2 red and 4 white •  B2 – 1 red and 2 white •  B3 – 5 red and 4 white A ball is randomly removed from one the bags. The bags were selected with probability: •  P(B1) = 1/3 •  P(B2) = 5/12 •  P(B3) = 1/4 What is the probability that the ball came from B1, given it is red?

Probability Theory

Conditional probability: Partition Theorem: Bayes’ Theorem:

P(A | B) = P(A∩B)P(B)

P(A) = P(A∩Bi )i∑ If the partition  

P(A | B) = P(B | A)P(A)P(B)

Bi A

Random Variables

A Random Variable is an object whose value is determined by chance, i.e. random events Maps elements of Ω onto real numbers, with corresponding probabilities as specified by P

Random Variables

A Random Variable is an object whose value is determined by chance, i.e. random events Maps elements of Ω onto real numbers, with corresponding probabilities as specified by P Formally, a Random Variable is a function:

X :Ω→R

Random Variables

A Random Variable is an object whose value is determined by chance, i.e. random events Maps elements of Ω onto real numbers, with corresponding probabilities as specified by P Formally, a Random Variable is a function: Probability that the random variable X adopts a particular value x:

P({w ∈Ω : X(w) = x})

X :Ω→R

Random Variables

A Random Variable is an object whose value is determined by chance, i.e. random events Maps elements of Ω onto real numbers, with corresponding probabilities as specified by P Formally, a Random Variable is a function: Probability that the random variable X adopts a particular value x:

P({w ∈Ω : X(w) = x})

P(X = x)Shorthand:

X :Ω→R

Random Variables

Example:

Ω = {heads, tails}X :Ω→ {0,1}

P(X = x) =P({heads}) x =1

P({tails}) x = 0

!

"##

$##

If the result is heads then WIN - X takes the value 1 If the result is tails then LOSE - X takes the value 0

P(X = x) =1 2 x ∈ {0,1}

Random Variables

Example: Win £20 on a six, nothing on four/five, lose £10 on one/two/three

Ω = {one, two, three, four, five, six}

X :Ω→ {−10,0, 20}

Random Variables

Example: Win £20 on a six, nothing on four/five, lose £10 on one/two/three

Ω = {one, two, three, four, five, six}

X :Ω→ {−10,0, 20}

P(X = x) =

P({six}) =1 6 x = 20

P({ four, five})=1 3 x = 0

P({one, two, three}) =1 2 x = −10

"

#

$$$

%

$$$

Note – we are considering the probabilities of events in

Probability Mass Functions

Given a random variable: The Probability Mass Function is defined as: Only for discrete random variables

X :Ω→ A

pX (x) = P(X = x)

Probability Mass Functions Example:

Win £20 on a six, nothing on four/five, lose £10 on one/two/three

pX (x) =

P({six}) =1 6 x = 20

P({ four, five})=1 3 x = 0

P({one, two, three}) =1 2 x = −10

"

#

$$$

%

$$$

Probability Mass Functions

Notable properties of Probability Mass Functions:

pX (x) ≥ 0

pX (x)x∈A∑ =1

Probability Mass Functions

Notable properties of Probability Mass Functions:

Interesting note: If p() is some function that has the above two properties, then it is the mass function of some random variable…

pX (x)x∈A∑ =1

pX (x) ≥ 0

Mean:

X :Ω→ AFor a random variable

E(X) = xpX (x)x∈A∑

Probability Mass Functions

Probability Mass Functions

Mean:

X :Ω→ AFor a random variable

E(X) = xpX (x)x∈A∑

Probability Mass Functions

Mean:

X :Ω→ AFor a random variable

E(X) = xpX (x)x∈A∑ 1

nxi

i=1

n

∑Compare with:

Probability Mass Functions

Mean:

X :Ω→ AFor a random variable

E(X) = xpX (x)x∈A∑

Median: pX (x)x≤m∑ ≥1 2 pX (x)

x≥m∑ ≥1 2any m such that: and

Probability Mass Functions

Mean:

X :Ω→ AFor a random variable

E(X) = xpX (x)x∈A∑

Median: any m such that: and

Mode: : most likely value argmaxx

pX (x)( )

pX (x)x≤m∑ ≥1 2 pX (x)

x≥m∑ ≥1 2

Common Discrete Distributions

Common Discrete Distributions

The Bernoulli Distribution:

p : success probability

X :Ω→ {0,1} pX (x) =p x =1

1− p x = 0

"

#$$

%$$

X ~ Bern(p)

Common Discrete Distributions

The Bernoulli Distribution:

p : success probability

X :Ω→ {0,1}

X ~ Bern(p)

Example:

X : {heads, tails}→ {0,1}

pX (x) =1 2 x ∈ {0,1}

X ~ Bern(1 2)Therefore

pX (x) =p x =1

1− p x = 0

"

#$$

%$$

Common Discrete Distributions

X ~ Bin(n, p)

pX (x) =nx!

"#$

%& px (1− p)n−xX :Ω→ {0,1,...,n}

The Binomial Distribution:

n : number of independent trials p : success probability

E(X) = np

Common Discrete Distributions

X ~ Bin(n, p)

pX (x) =nx!

"#$

%& px (1− p)n−xX :Ω→ {0,1,...,n}

The Binomial Distribution:

n : number of independent trials p : success probability

pX (x) : probability of getting x successes out of n trials : probability of x successes : probability of (n-x) failures : number of ways to achieve x successes and (n-x) failures (Binomial coefficient)

px

(1− p)n−x

nx!

"#$

%&=

n!x!(n− x)!

E(X) = np

Common Discrete Distributions

X ~ Bin(n, p)

pX (x) =nx!

"#$

%& px (1− p)n−xX :Ω→ {0,1,...,n}

The Binomial Distribution:

n : number of independent trials p : success probability

n =1 : pX (x) = px (1− p)1−x =p x =1

1− p x = 0

"#$

%$

X ~ Bin(1, p) ⇔ X ~ Bern(p)

E(X) = np

Common Discrete Distributions

X ~ Bin(n, p)

pX (x) =nx!

"#$

%& px (1− p)n−xX :Ω→ {0,1,...,n}

The Binomial Distribution:

n : number of independent trials p : success probability

E(X) = np

n =10p = 0.5

Common Discrete Distributions

X ~ Bin(n, p)

pX (x) =nx!

"#$

%& px (1− p)n−xX :Ω→ {0,1,...,n}

The Binomial Distribution:

n : number of independent trials p : success probability

E(X) = np

n =100p = 0.5

Common Discrete Distributions

X ~ Bin(n, p)

pX (x) =nx!

"#$

%& px (1− p)n−xX :Ω→ {0,1,...,n}

The Binomial Distribution:

n : number of independent trials p : success probability

E(X) = np

n =100p = 0.8

Common Discrete Distributions

Example:

Number of heads in n fair coin toss trials

Ω = {heads :heads,heads : tails, tails :heads, tails : tails}n = 2

Ω = 2nIn general:

X :Ω→ {0,1,...,n}

Common Discrete Distributions

Example:

X ~ Bin(n,1 2)

Number of heads in n fair coin toss trials

Ω = {heads :heads,heads : tails, tails :heads, tails : tails}n = 2

Ω = 2n

pX (x) =nx!

"#$

%&0.5n

In general:

E(X) = n 2

Notice:

X :Ω→ {0,1,...,n}

Common Discrete Distributions

X ~ Pois(λ)

pX (x) =λ xe−λ

x!X :Ω→ {0,1,...}

The Poisson Distribution: E(X) = λ

λ : average number of counts (controls rarity of events)

Used to model the number of occurrences of an event that occur within a particular interval of time and/or space

Common Discrete Distributions

•  Want to know the distribution of the number of occurrences of an event Binomial?

•  However, don’t know how many trials are performed – could be infinite!

•  But we do know the average rate of occurrence:

The Poisson Distribution:

⇒

E(X) = λ

X ~ Bin(n, p) ⇒ E(X) = np⇒ λ = np

⇒ p = λn

Common Discrete Distributions

Binomial: pX (x) =n!

x!(n− x)!px (1− p)n−x

p = λn

⇒ pX (x) =n!

x!(n− x)!λn#

$%

&

'(x

1− λn

#

$%

&

'(n−x

Common Discrete Distributions

Binomial: pX (x) =n!

x!(n− x)!px (1− p)n−x

p = λn

⇒ pX (x) =n!

x!(n− x)!λn#

$%

&

'(x

1− λn

#

$%

&

'(n−x

pX (x) =λ x

x!n!

nx (n− x)!

1− λn

#

$%

&

'(n

1− λn

#

$%

&

'(x

pX (x) =n!

x!(n− x)!px (1− p)n−x

p = λn

⇒ pX (x) =n!

x!(n− x)!λn#

$%

&

'(x

1− λn

#

$%

&

'(n−x

pX (x) =λ x

x!n!

nx (n− x)!

1− λn

#

$%

&

'(n

1− λn

#

$%

&

'(x

Common Discrete Distributions

Binomial:

1  

1  as n→∞

pX (x) =n!

x!(n− x)!px (1− p)n−x

p = λn

⇒ pX (x) =n!

x!(n− x)!λn#

$%

&

'(x

1− λn

#

$%

&

'(n−x

pX (x) =λ x

x!n!

nx (n− x)!

1− λn

#

$%

&

'(n

1− λn

#

$%

&

'(x

Common Discrete Distributions

Binomial:

1  

1  as n→∞

pX (x) = Limn→∞

λ x

x!1− λ

n$

%&

'

()n$

%&&

'

())

pX (x) =n!

x!(n− x)!px (1− p)n−x

p = λn

⇒ pX (x) =n!

x!(n− x)!λn#

$%

&

'(x

1− λn

#

$%

&

'(n−x

pX (x) =λ x

x!n!

nx (n− x)!

1− λn

#

$%

&

'(n

1− λn

#

$%

&

'(x

Common Discrete Distributions

pX (x) = Limn→∞

λ x

x!1− λ

n$

%&

'

()n$

%&&

'

()) =

λ x

x!e−λ

Binomial:

1  

1  as n→∞

☐

This is referred to as the: •  “Poisson Limit Theorem” •  “Poisson Approximation to the Binomial” •  “Law of Rare Events”

Common Discrete Distributions

The Poisson distribution is the Binomial distribution as If then If n is large and p is small then the Binomial distribution can be approximated using the Poisson distribution

n→∞

Xn ~ Bin(n, p) Xnd! →! Pois(np)

λ: n→∞ ⇒ p→ 0fixed

Poisson is often more computationally convenient than Binomial

References

Countless books + online resources! Probability theory and distributions: •  Grimmett and Stirzker (2001) Probability and Random

Processes. Oxford University Press. General comprehensive introduction to (almost) everything mathematics (including a bit of probability theory): •  Garrity (2002) All the mathematics you missed: but need to

know for graduate school. Cambridge University Press.